- Sesquilinear form
In
mathematics , a sesquilinear form on acomplex vector space "V" is a map "V" × "V" → C that is linear in one argument andantilinear in the other. The name originates from thenumerical prefix meaning "one and a half". Compare with abilinear form , which is linear in both arguments; although it should be noted that many authors, especially when working solely in a complex setting, refer to sesquilinear forms as bilinear forms.A motivating example is the
inner product on a complex vector space, which is not bilinear, but instead sesquilinear. See geometric motivation below.Definition and conventions
Conventions differ as to which argument should be linear. We take the first to be conjugate-linear and the second to be linear. This is the convention used by essentially all physicists and originates in Dirac's
bra-ket notation inquantum mechanics . The opposite convention is perhaps more common in mathematics but is not universal.Specifically a map φ : "V" × "V" → C is sesquilinear if:for all "x,y,z,w" ∈ "V" and all "a", "b" ∈ C.
A sesquilinear form can also be viewed as a
bilinear map :where is thecomplex conjugate vector space to "V". By the universal property oftensor product s these are in one-to-one correspondence with (complex) linear maps:For a fixed "z" in "V" the map is a
linear functional on "V" (i.e. an element of thedual space "V"*). Likewise, the map is aconjugate-linear functional on "V".Given any sesquilinear form φ on "V" we can define a second sesquilinear form ψ via the
conjugate transpose ::In general, ψ and φ will be different. If they are the same then φ is said to be "Hermitian". If they are negatives of one another, then φ is said to be "skew-Hermitian". Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.Geometric motivation
Bilinear forms are to squaring (), what sesquilinear forms are to
Euclidean norm ().The norm associated to a sesquilinear form is invariant under multiplication by the complex circle (complex numbers of unit norm), while the norm associated to a bilinear form is
equivariant (with respect to squaring). Bilinear forms are "algebraically " more natural, while sesquilinear forms are "geometrically" more natural.If "B" is a bilinear form on a complex vector space and is the associated norm, then .
By contrast, if "S" is a sesquilinear form on a complex vector space and is the associated norm, then .
Hermitian form
:"The term Hermitian form may also refer to a different concept than that explained below: it may refer to a certain
differential form on aHermitian manifold ."A Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form "h" : "V" × "V" → C such that:The standard Hermitian form on C"n" is given by:More generally, the
inner product on anyHilbert space is a Hermitian form.A vector space with a Hermitian form ("V","h") is called a Hermitian space.
If "V" is a finite-dimensional space, then relative to any basis {"e""i"} of "V", a Hermitian form is represented by a
Hermitian matrix H::The components of H are given by "H""ij" = "h"("e""i", "e""j").The
quadratic form associated to a Hermitian form:"Q"("z") = "h"("z","z")is always real. Actually one can show that a sesquilinear form is Hermitianiff the associated quadratic form is real for all "z" ∈ "V".Skew-Hermitian form
A skew-Hermitian form (also called an antisymmetric sesquilinear form), is a sesquilinear form ε : "V" × "V" → C such that:Every skew-Hermitian form can be written as "i" times a Hermitian form.
If "V" is a finite-dimensional space, then relative to any basis {"e""i"} of "V", a skew-Hermitian form is represented by a
skew-Hermitian matrix A::The quadratic form associated to a skew-Hermitian form:"Q"("z") = ε("z","z")is always pure imaginary.
Generalization: over a *-ring
A sesquilinear form and a Hermitian form can be defined over any
*-ring , and the examples of symmetric bilinear forms, skew-symmetric bilinear forms, Hermitian forms, and skew-Hermitian forms, are all Hermitian forms for various involutions.Particularly in
L-theory , one also sees the term ε-symmetric form, where , to refer to both symmetric and skew-symmetric forms.References
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